The more we know the more we notice.

Posts Tagged ‘trig’

How far away are the clouds on the horizon? The answer may surprise you.

First, where is your horizon? On the plains, it is pretty low, maybe a few degrees above the horizontal, but when you live in town or in hills, your lowest view of the sky may be quite high. On a mountain, your horizon is as low as possible. But what is interesting is that except from a mountaintop, your land view doesn’t go much more than six miles, while your sky view easily goes twenty miles even with simple cumulus. With higher clouds, it may reach 40 miles or more, less in town, more in the plains.

To understand this, what we want to consider is the little thing that mathematicians call the tangent, which, for this discussion, is the ratio between the height of a cloud and its horizontal distance from us. When you look up and over at a 45 angle, (point exactly halfway between straight up and straight over) the cloud you see is directly above the land that is as far away as the cloud is high. So if the clouds are one mile high, the cloud you see at a 45 degree angle is above the land that is 1 mile away. If the clouds are two miles high, then the cloud you see at a 45 degree angle is two miles away. If the clouds are 6 miles high (as those angel-hair cirrus tend to be) then what you see at 45 degrees actually stands over the land that is as far as your eye can normally see, six miles away; everything else in the sky below that is standing over land spaces that you cannot see except from a highrise, a hill, or a mountaintop.

Now don’t run away, you math-o-phobes! This is too cool! I want to give you a few more angles because they show something very interesting:

If you look out at a 30 degree angle, the clouds you see are about twice as far away as they are high. If you look out at 9 degrees, they are six times as far away; for 8 degrees it’s about seven times as far, and if you look out at 3 degrees above the horizon (and if the world were flat) it would be almost 20 times as far — that is, the clouds you would see just above the horizon would be 20 times as far away as high. Even though the world is not flat, however, the point is that as you look lower in the sky, the clouds you see are standing above increasingly more distant spaces. You may imagine that if the clouds at 45 degrees are one mile away, then the clouds at 90 degrees (on the horizon) will be two miles away. No, not at all. They are more likely 20 miles away and may be 60 miles away.

That is the main reason why clouds seem all stacked up at the horizon. You are seeing much greater distances into the sky than out on the land. I have seen the work of an artist whose clouds lacked reality for just this reason. She was trying to draw the clouds that she could see “above” her little field, but in real life, we see clouds from many miles beyond the fields in front of us, and the distant ones look different from the ones that really are above the local field.

We see only a few miles of land, but we see 20, 30, or perhaps 50 miles of sky. Think about it! We will come back to this.